Editing Antiprism (section)

====Number of uniform crossed antiprisms====
If the notation {{math|(''p''/''q'')}} is used for an antiprism, then for {{math|''q'' > ''p''/2}} the antiprism is crossed (by definition) and for {{math|''q'' < ''p''/2}} is not. In this section all antiprisms are assumed to be non-degenerate, i.e. {{math|''p'' ≥ 3}}, {{math|''q'' ≠ ''p''/2}}. Also, the condition {{math|(''p'',''q'') {{=}} 1}} ({{mvar|p}} and {{mvar|q}} are relatively prime) holds, as compounds are excluded from counting. The number of uniform crossed antiprisms for fixed {{mvar|p}} can be determined using simple inequalities. The condition on possible {{mvar|q}} is

: {{math|{{sfrac|''p''|2}} < ''q'' < {{sfrac|2|3}} ''p''}} and {{math|1=(''p'',''q'') = 1.}}

Examples:
* {{mvar|''p''}} = 3: {{sfrac|''p''|2}} = 1.5 and {{sfrac|2|3}} {{mvar|p}} = 2, so 2 ≤ {{mvar|q}} ≤  1 – a uniform triangular crossed antiprism does not exist.
* {{mvar|''p''}} = 5: {{sfrac|''p''|2}} = 2.5 and {{sfrac|2|3}} {{mvar|p}} = {{sfrac|10|3}} = {{sfrac|3|1|3}}, so 3 ≤ {{mvar|q}} ≤ 3 – one antiprism of the type (5/3) can be uniform.
* {{mvar|''p''}} = 29: {{sfrac|''p''|2}} = 14.5 and {{sfrac|2|3}} {{mvar|p}} = {{sfrac|58|3}} = {{sfrac|19|1|3}}, 15 ≤ {{mvar|q}} ≤ 19 – there are five possibilities shown in the rightmost column, below the (29/1) convex antiprism, on the image above.
* {{mvar|''p''}} = 15: {{sfrac|''p''|2}} = 7.5 and {{sfrac|2|3}} {{mvar|p}} = 10, 8 ≤ {{mvar|q}} ≤ 9 – antiprism with {{mvar|q}} = 8 is a solution, but {{mvar|q}} = 9 must be rejected, as (15,9) = 3 and {{sfrac|15|9}} = {{sfrac|5|3}}. The antiprism (15/9) is a compound of three antiprisms (5/3). Since 9 satisfies the inequalities, the compound can be uniform, and if it is, then its parts must be. Indeed, the antiprism (5/3) can be uniform by example 2.

In the first column of the following table, the symbols are Schoenflies, Coxeter, and orbifold notation, in this order.
{| class="wikitable mw-collapsible mw-collapsed" style="text-align:center"
|+ class="nowrap"| Star ({{math|''p''/''q''}})-antiprisms by symmetry, for {{math|''p'' ≤ 12}}
! [[List of spherical symmetry groups|Symmetry group]]
! colspan=4 | Uniform stars
! Right stars
|-
! {{math|D<sub>3h</sub><br>[2,3]<br>(2*3)}}
| colspan=4 |
| [[Image:Crossed triangular antiprism.svg|64px]]<br>3.3/2.3.3<br>[[Crossed triangular antiprism]]
|-
! {{math|D<sub>4d</sub><BR>[2<sup>+</sup>,8]<BR>(2*4)}}
| colspan=4 |
| [[Image:Crossed square antiprism.png|64px]]<BR>3.3/2.3.4<br>[[Crossed square antiprism]]
|-
! {{math|D<sub>5h</sub><BR>[2,5]<BR>(*225)}}
| [[Image:Pentagrammic antiprism.png|64px]]<BR>3.3.3.5/2<br>[[Pentagrammic antiprism]]
| colspan=3 |
| [[Image:Crossed pentagonal antiprism.png|64px]]<BR>3.3/2.3.5<br>[[Crossed pentagonal antiprism]]
|-
! {{math|D<sub>5d</sub><BR>[2<sup>+</sup>,10]<BR>(2*5)}}
| [[Image:Pentagrammic crossed antiprism.png|64px]]<BR>3.3.3.5/3<br>[[Pentagrammic crossed-antiprism]]
|-
! {{math|D<sub>6d</sub><BR>[2<sup>+</sup>,12]<BR>(2*6)}}
| colspan=4 |
| [[Image:Crossed hexagonal antiprism.png|64px]]<BR>3.3/2.3.6<br>[[Crossed hexagonal antiprism]]
|-
! {{math|D<sub>7h</sub><BR>[2,7]<BR>(*227)}}
| [[Image:Antiprism 7-2.png|64px]]<BR>3.3.3.7/2<br>Heptagrammic antiprism (7/2)
| [[Image:Antiprism 7-4.png|64px]]<BR>3.3.3.7/4<br>Heptagrammic crossed antiprism (7/4)
|-
! {{math|D<sub>7d</sub><BR>[2<sup>+</sup>,14]<BR>(2*7)}}
| [[Image:Antiprism 7-3.png|64px]]<BR>3.3.3.7/3<br>Heptagrammic antiprism (7/3)
|-
! {{math|D<sub>8d</sub><BR>[2<sup>+</sup>,16]<BR>(2*8)}}
| [[Image:Antiprism 8-3.png|64px]]<BR>3.3.3.8/3<br>[[Octagrammic antiprism]]
| [[Image:Antiprism 8-5.png|64px]]<BR>3.3.3.8/5<br>[[Octagrammic crossed-antiprism]]
|-
! {{math|D<sub>9h</sub><BR>[2,9]<BR>(*229)}}
| [[Image:Antiprism 9-2.png|64px]]<BR>3.3.3.9/2<br>[[Enneagrammic antiprism (9/2)]]
| [[Image:Antiprism 9-4.png|64px]]<BR>3.3.3.9/4<br>[[Enneagrammic antiprism (9/4)]]
|-
! {{math|D<sub>9d</sub><BR>[2<sup>+</sup>,18]<BR>(2*9)}}
| [[Image:Antiprism 9-5.png|64px]]<BR>3.3.3.9/5<br>[[Enneagrammic crossed-antiprism]]
|-
! {{math|D<sub>10d</sub><BR>[2<sup>+</sup>,20]<BR>(2*10)}}
| [[Image:Antiprism 10-3.png|64px]]<BR>3.3.3.10/3<br>[[Decagrammic antiprism]]
|-
! {{math|D<sub>11h</sub><BR>[2,11]<BR>(*2.2.11)}}
| [[Image:Antiprism 11-2.png|64px]]<BR>3.3.3.11/2<br>Undecagrammic (11/2)
| [[Image:Antiprism 11-4.png|64px]]<BR>3.3.3.11/4<br>Undecagrammic (11/4)
| [[Image:Antiprism 11-6.png|64px]]<BR>3.3.3.11/6<br>Undecagrammic crossed (11/6)
|-
! {{math|D<sub>11d</sub><BR>[2<sup>+</sup>,22]<BR>(2*11)}}
| [[Image:Antiprism 11-3.png|64px]]<BR>3.3.3.11/3<br>Undecagrammic (11/3)
| [[Image:Antiprism 11-5.png|64px]]<BR>3.3.3.11/5<br>Undecagrammic (11/5)
| [[Image:Antiprism 11-7.png|64px]]<BR>3.3.3.11/7<br>Undecagrammic crossed (11/7)
|-
! {{math|D<sub>12d</sub><BR>[2<sup>+</sup>,24]<BR>(2*12)}}
| [[Image:Antiprism 12-5.png|64px]]<BR>3.3.3.12/5<br>Dodecagrammic
| [[Image:Antiprism 12-7.png|64px]]<BR>3.3.3.12/7<br>Dodecagrammic crossed
|-
! ...
| ...
|}